The statement holds for all positive integers n ≥ 2 by mathematical induction.
How did we arrive at this assertion?We will use mathematical induction to prove that for all integers n ≥ 2,
(2C2) + (3C2) + ... + (nC2) = (n+1)C2.
Base Case:
For n=2, we have:
(2C2) = 1 and (2+1)C2 = 3C2 = 3/2
So the statement holds for the base case n=2.
Inductive Hypothesis:
Assume that the statement is true for some arbitrary positive integer k, i.e.,
(2C2) + (3C2) + ... + (kC2) = (k+1)C2
Inductive Step:
We must prove that the statement is also true for k+1, i.e.,
(2C2) + (3C2) + ... + ((k+1)C2) = (k+2)C2
Expanding the left-hand side of this equation, we have:
(2C2) + (3C2) + ... + ((k+1)C2) = [(2C2) + (3C2) + ... + (kC2)] + (k+1)C2
Using the inductive hypothesis, we know that:
(2C2) + (3C2) + ... + (kC2) = (k+1)C2
So we can substitute this expression to get:
(2C2) + (3C2) + ... + ((k+1)C2) = (k+1)C2 + (k+1)C2
Using the identity (n+1)C2 = nC2 + n + 1, we can simplify the right-hand side of this equation to get:
(2C2) + (3C2) + ... + ((k+1)C2) = 2(k+1)C2
Now, using the identity (n+1)C2 = nC2 + n + 1 again, we can simplify the right-hand side to get:
2(k+1)C2 = (k+1)(k+2)C2
So, we have:
(2C2) + (3C2) + ... + ((k+1)C2) = (k+1)(k+2)C2
This is the same as the right-hand side of the equation we wanted to prove. Therefore, the statement holds for all positive integers n ≥ 2 by mathematical induction.
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using the washer method, determine the volume of a solid formed by revolving the region in the first quadrant bounded on the left by the circle , on the right by the line , and above by the line about the -axis.
The volume of a solid formed by revolving the region in the first quadrant bounded on the left by the circle , on the right by the line , and above by the line about the x-axis is (47/3)π.
To use the washer method, we need to consider the solid as a stack of thin washers, each with a hole in the center. We will integrate along the axis of revolution (in this case, the x-axis).
First, let's sketch the region we're revolving:
The circle has equation x^2 + y^2 = 1, so its radius is r = 1 when y = 0. The line has equation x = 4, so its distance from the axis of revolution is R = 4. The region we're revolving is bounded by y = 0 and y = √(1 - x^2), which we can rewrite as x = √(1 - y^2).
We can now set up the integral for the volume of the solid:
V = [tex]\int\limits^1_0[/tex] π(R^2 - r^2) dy
= [tex]\int\limits^1_0[/tex] π(4^2 - x^2) dy
= π[tex]\int\limits^1_0[/tex] (16 - y^2) dy
= π [16y - (1/3)y^3] |₀¹
= π (16 - 1/3)
= (47/3)π
Therefore, the volume of the solid is (47/3)π cubic units.
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Complete question is:
Using the washer method, determine the volume of a solid formed by revolving the region in the first quadrant bounded on the left by the circle x^2+y^2=1, on the right by the line x = 4, and above by the line x=4 about the x-axis.
John buys 2 shirts for $7.89, he also buys a pair of shoes for $15.If he has a sales tax of 8.9%. What would be his final cost?
Answer:
$33.52
Step-by-step explanation:
7.89+7.89+15= 30.78
30.78x 0.089=2.73942 rounded off to the nearest hundredth which is 2.74
30.78+2.74= $33.52
Find x from two right triangles
The length of x is 7.
Describe Triangles?In geometry, a triangle is a three-sided polygon made up of three line segments that join at three points called vertices. Triangles are one of the most basic and important shapes in geometry, and they are used in many areas of mathematics, science, and engineering.
Using the property of a median, we know that BD divides AC into two equal parts. Let's call the length of CD as y.
So, we have:
AC = AD + CD
x = 3 + y
Using the Pythagorean Theorem, we can find the length of BD:
BD² = AB² + AD² - 2(AB)(AD)(cos(B))
BD² = 4² + 3² - 2(4)(3)(cos(B))
BD² = 25 - 24cos(B)
BD = √(25 - 24cos(B))
Since BD is a median, we know that it divides BC into two equal parts. Let's call the length of BD as z.
So, we have:
BC = 2z
8 = 2z
z = 4
Now, using the property of medians again, we know that:
4² + y² = z²
16 + y² = 16
y² = 0
y = 0
Therefore, CD is also 4, and:
x = 3 + y
x = 3 + 4
x = 7
So, the length of AC is 7.
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50 POINTS PLEASE HELP SOLVE
What is the true solution to the logarithmic equation?
log₂[log₂ (√4x)] = 1
O x=-4
O x=0
O x=2
Ox=4
Step-by-step explanation:
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The only solution to the given logarithmic equation is x = 4.
What is Logarithmic?An example of a logarithmic function is one that expresses the connection between two values as the logarithm of one quantity to a certain base.
A logarithmic function's basic form is:
f(x)=log base b (x)
where x is the function's input, b is the logarithm's base, and f(x) is the function's value.
We can solve the logarithmic equation step by step as follows:
log₂[log₂ (√4x)] = 1
First, we simplify the expression inside the logarithm:
log₂[log₂ (√4x)] = 1
log₂[log₂ (2√x)] = 1
Next, we use the definition of logarithms to rewrite the equation in exponential form:
log₂ (2√x) = 2¹ = 2
2√x = 2² = 4
√x = 2
x = 4
Thus, the solution is x=4.
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Consider a one-sample two-sided z-test for a population proportion. Given that conditions for inference are met, which of the following is closest to the p-value for a test statistic of z=−1.86 ?
Answer: 0.0628
Step-by-step explanation:
The closest answer to the p-value for a test statistic of z = -1.86 is 0.0614.
To calculate the p-value for a one-sample two-sided z-test on a population proportion, we must first calculate the area under the standard normal distribution curve that corresponds to the test statistic z = -1.86.
Because this is a two-sided test, we must find the area of the distribution in both tails that is more extreme than the test statistic. We can do this by using a standard normal distribution table or calculator, or by exploiting the distribution's symmetry.
We can calculate the area to the left of z = -1.86 using a standard normal distribution table or calculator. Because this is a two-sided test, the area in the distribution's right tail is also 0.0307.As a result, the p-value for this test statistic is as follows:
2(0.0307) = 0.0614 p-value
So 0.0614 is the closest answer to the p-value for a test statistic of z = -1.86.
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Determine which variable is the likely explanatory variable and which is the likely response variable.-The explanatory variable is the miles per gallon and the response variable is the weight.-The explanatory variable is the weight and the response variable is the miles per gallon.
For the first statement, the likely explanatory variable is the miles per gallon and the likely response variable is the weight.
For the second statement, the likely explanatory variable is the weight and the likely response variable is the miles per gallon.
In statistical analysis, an explanatory variable is a variable that is presumed to have an effect on another variable. It is also known as an independent variable. On the other hand, a response variable is the variable that is presumed to be affected by the explanatory variable. It is also known as a dependent variable.
In the first statement, it is stated that the explanatory variable is the miles per gallon, which means that we are looking at how the fuel efficiency of a vehicle affects its weight. Therefore, the weight is the likely response variable, as it is presumed to be affected by the miles per gallon.
In the second statement, it is stated that the explanatory variable is the weight, which means that we are looking at how the weight of a vehicle affects its fuel efficiency. Therefore, the miles per gallon is the likely response variable, as it is presumed to be affected by the weight.
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Vera rents bicycles to tourists. She recorded the height (in \text{cm}cmstart text, c, m, end text) of each customer and the frame size (in \text{cm}cmstart text, c, m, end text) of the bicycle that customer rented.
After plotting her results, Vera noticed that the relationship between the two variables was fairly linear, so she used the data to calculate the following least squares regression equation for predicting bicycle frame size from the height of the customer:
\hat{y} = \dfrac{1}{3}+\dfrac{1}{3}x
y
^
=
3
1
+
3
1
xy, with, hat, on top, equals, start fraction, 1, divided by, 3, end fraction, plus, start fraction, 1, divided by, 3, end fraction, x
What is the residual of a customer with a height of 155\,\text{cm}155cm155, start text, c, m, end text who rents a bike with a 51\,\text{cm}51cm51, start text, c, m, end text frame?
The residual of a customer with a height of 155 cm who rents a bike with a 51 cm frame is -1.
What is Residual?The discrepancy between a response variable's observed value and its expected value as predicted by the regression line.
We have the Equation: y'=1/3x + 1/3.
The regression equation is given as:
y'=(1/3)x + (1/3)
Since the height is given as 155cm, x=155 cm
The predicted frame size,
y'=(1/3)x + (1/3)
y'=(1/3) × 155+ (1/3)
= 51 2/3 + 1/3
= 52
So, Residual = Observed y- predicted y
= 51-52
= -1
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Answer:
Step-by-step explanation:
Please like if this helped :)
You will need a pencil and paper.
Shalonda is deciding what to wear. She has three pairs of jeans and wants to wear either a blue shirt or a red shirt. How many different combinations are possible?
5 combinations
6 combinations
7 combinations
8 combinations
Answer:
6 combinations
Step-by-step explanation:
3 pairs of Jeans & 2 shirts
3 x 2 = 6
6 combinations
Two fractions are equal. They also have the same denominator. What must be true of the numerators of the fractions? Explain.
Answer: The numerators must be the same for the fractions to be equal.
The denominator of a fraction is the bottom number. Take a look at these two fractions.
¾ & ¾
The numbers at the bottom are the same. If I eat ¾ of an apple and you eat ¾ of an apple, we both eat the same amount.
Now look at these fractions.
⅗ & ¾
They both have the same top number (numerator), but they have different denominators. If I eat ¾ of an apple and you eat ⅗ of an apple, we both do not eat the same amount.
So if both the numerator and denominator are the same they are equal.
This means that the numerators must be the same for the fractions to be equal.
I hope this helps & Good Luck <3 !!!
Can I get help please
Answer:
B. Vertex
Step-by-step explanation:
The midpoint between the two x-intercepts of a parabola will provide the x-coordinate of the vertex. Line of symmetry is drawn along this particular x-coordinate. This x-coordinate is substituted into the equation of the graph to find its corresponding y-coordinate.
the graph is provided for questions a-h
The probability that a randomly selected medical student who took the test had a total score that was less than 480 is 0.49, which means that approximately 49% of medical students had a score lower than 480.
What is probability?Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event.
here, we have,
To calculate the probability that a randomly selected medical student who took the test had a total score that was less than 480, we need to start by looking at the data given.
In this case, the data tells us that the mean score was 500 and the standard deviation was 50.
This means that the scores are normally distributed, which means that the probability of finding a score lower than 480 can be calculated using the normal distribution formula.
We can use this formula to calculate the area under the curve (or probability) below the score of 480.
The result of this calculation is 0.49, which means that approximately 49% of medical students had a score lower than 480.
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Peter travels at an average speed of 98km/h for 3 hours.he then travels at an average speed of 105km/h for 2 Hours
Answer:
Assuming that Peter travels at a constant speed, then he would have travelled a total distance of (98 * 3) + (105 * 2) = 531 km.
: A major corporation is building a 4325 acre complex of homes, offices, stores, schools, and churches in the rural community of Glen Cove. As a result of this development, the planners have estimated that Glen Clove's population (in thousands) t yr from now will be given by the following function.P(t) = 35t^2 + 125t + 200/t^2 + 4t + 40(a) What is the current population of Glen Cove? (b) What will be the population in the long run?
The current population of Glen Cove is 5,000 people and in the long-run, the population of Glen Cove will approach 35,000 people.
Corporation:A corporation may be a corporate entity whose shareholders elect a board of directors to supervise its operations. the dissimilarity between the business and an organization is that an organization may be a sort of business that's suitable for smaller businesses and organizations, whereas a corporation may be a kind of business that's appropriate for larger enterprises and entities.
The population in Glen Clove's population that is t years from now will be
P(t) = [tex]\frac{35t^2 + 125t + 200}{t^2 + 4t + 40}[/tex]
The current population can be calculated by substituting t = 0 in P(t).
So,
P(0) = [tex]\frac{35(0)^2 + 125(0) + 200}{(0)^2 + 4(0) + 40}[/tex]
P(0) = 5
Thus, the current population of Glen Clove is 5 thousand.
(b)For the long run, t tends to infinity. So, evaluate the infinite limit for the given function as follows:
[tex]\lim_{t \to \infty} P(t)= \lim_{t \to \infty}P(t)\frac{35t^2 + 125t + 200}{t^2 + 4t + 40}[/tex]
=[tex]\lim_{t \to \infty}\frac{t^2(35+\frac{125}{t} +\frac{200}{t} )}{t^2(1+\frac{4}{t} +\frac{40}{t^2} )}[/tex]
= [tex]\lim_{t \to \infty}\frac{(35+\frac{125}{t} +\frac{200}{t} )}{1+(\frac{4}{t} +\frac{40}{t^2} )}[/tex]
Now, solve further by putting the limits as follows:
As t approaches infinity, the terms in the numerator approach 0, while the terms in the denominator approach 1, so the limit of P(t) as t approaches infinity is:
lim t -> infinity P(t) = 35 / 1 = 35
So, in the long-run, the population of Glen Cove will approach 35,000 people.
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need help with my misisng he asap plss 30 points
Answer:
y=4
x=3
Step-by-step explanation:
2y-4=y
y=4
4x=x+9
3x=9
x=3
consider the discrete random variable x given in the table below calculate the mean variance and standard deviation of X.
Answer:
Expected value E(x)
= 2 x 0.11 + 11 x 0.09 + 12 x 0.08 + 13 x 0.58 + 19 x 0.14
= 12.37
Variance
= E( X^2 ) - (E(X))^2
= (2^2 x 0.11 + 11^2 x 0.09 + 12^2 x 0.08 + 13^2 x 0.58 + 19^2 x 0.14) - (12.37)^2
= 171.41 - 153.0169
= 18.393 (cor to 3 dp)
Standard deviation
= [tex]\sqrt{variance}[/tex]
=[tex]\sqrt{18.3931}[/tex]
= 4.289 (cor to 3 dp)
Mean should be ( [tex]\frac{2+11+12+13+19}{5}[/tex] ) = 11.4 but i am not very sure
Mean, variance and the standard deviation of the discrete random variable are 12.37, 18.3931 and 4.2887 respectively.
What is Expected Value in Probability?Expected value in probability is defined as the sum of the product of each of the variable and probability corresponding to the variable.
Expected value is same as the mean of a discrete random variable.
Mean, μ = Σ x. P(x)
= (2 × 0.11) + (11 × 0.09) + (12 × 0.08) + (13 × 0.58) + (19 × 0.14)
= 12.37
Variance, σ² = Σx² P(x) - μ²
= [(2² × 0.11) + (11² × 0.09) + (12² × 0.08) + (13² × 0.58) + (19² ×
0.14)]- [ 12.37²]
= 18.3931
Standard Deviation, σ = √Variance
= 4.2887
Hence the mean is 12.37, Variance is 18.3931 and Standard deviation is 4.2887.
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The existing entries in the following table show the maximum amount of goals scored and the amount of shots saved that Mia Hamm and Alyssa Naeher could make as soccer players during a year.
Goals Scored
Amount of shots saved as a goalie
Mia Hamm
50
30
Alyssa Naeher
20
120
Please answer the following questions below:
a. Does either Mia or Hamm have an absolute advantage in goals scored and amounts?
b. Calculate the opportunity cost of goals scoring 1 goal for Mia and Alyssa.
c. Calculate the opportunity cost of saving 1 shot as a goalie for Mia and Alyssa.
d. Who has the comparative advantage in scoring goals? Who has the comparative advantage in saving shots as a goalie? Why?
Please show all your work and calculations.
a. Yes, Mia Hamm has an absolute advantage in both scoring goals and saving shots as a goalie,
b. The opportunity cost of scoring 1 goal for Mia is 0.6 shots
c. The opportunity cost of saving 1 shot as a goalie for Mia is 1.67 goals . The opportunity cost of saving 1 shot as a goalie for Alyssa is 0.17 goals scored.
d. Mia has the comparative advantage in scoring goals, since her opportunity cost of scoring 1 goal is lower than Alyssa's. Alyssa has the comparative advantage in saving shots as a goalie, since her opportunity cost of saving 1 shot is lower than Mia's
How do we calculate?b. The opportunity cost of scoring 1 goal for Mia is 0.6 shots saved as a goalie which is calculated as
30 shots/50 goals = 0.6).
The opportunity cost of scoring 1 goal for Alyssa is 6 shots saved as a goalie which is calculate as
opportunity cost 120 shots/20 goals = 6.
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Question 12
A homeowner is planning to build a concrete sidewalk that will require 1 3/4 cubic yards of concrete.
She also will build a small patio requiring 2 1/3 cubic yards of concrete. The two projects are in the
backyard, so the concrete will be carried in a wheelbarrow. The wheelbarrow will comfortably hold
1/6 of cubic yard of concrete. How many wheelbarrow loads will be required for the entire job?
The total number of wheelbarrow loads required to complete both the projects 25.
What are Fractions?
Any number of equal parts is represented by a fraction, which also represents a portion of a whole. In ordinary English, a fraction refers to the number of parts of a specific size.
The combined volume of concrete needed for the patio and walkway is 1 3/4 + 2 1/3 = 7/4 + 7/3 = 21/12 + 28/12 = 49/12 cubic yards
The entire volume of concrete can be divided by the capacity of a wheelbarrow to determine the necessary number of wheelbarrow loads: (49/12 )/( 1/6) = 49/12* 6/1 = 49/2 = 24.5
We round up to 25 because we can't have half a wheelbarrow load. The homeowner will therefore require 25 wheelbarrow loads to finish both projects.
Hence, The total number of wheelbarrow loads required to complete both the projects 25.
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Identify the base: "Automobiles are 97% cleaner than they were 30 years ago
(97x/100) represents the total number of cleaner automobiles.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.
Given is that automobiles are 97% cleaner than they were 30 years ago.
Assume that the total number of automobiles 30 years ago were {x}. Then, the total number of clean automobiles will be -
97% of {x} = 97x/100
Therefore, (97x/100) represents the total number of cleaner automobiles.
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The manager of a furniture factory finds that it costs $2,200 to manufacture 100 chairs in one day and $4,800 to produce 300 chairs in one day.
a) Express the cost as a function of the number of chairs produced, assuming that it is linear.
b) Sketch the graph of the function obtained in part (a).
c) What is the slope of the graph and what does it represent?
d) What is the y-intercept of the graph and what does it represent?
a) C(x) = 0.02x + 2200, b) A straight line with positive slope, c) Slope is 0.02, represents cost per chair, d) Y-intercept is 2200, represents fixed cost.
a) The cost C(x) can be expressed as a linear function of the number of chairs x produced. We can solve for the slope and y-intercept using the two points (100, 2200) and (300, 4800). First, calculate the slope of the line using the formula (y2 - y1)/(x2 - x1). This gives us (4800 - 2200)/(300 - 100) = 0.02. Then, we can use the slope and one of the points to solve for the y-intercept using the equation y = mx + b, where m is the slope and b is the y-intercept. Substituting the values from the first point (100, 2200) into the equation gives us 2200 = 0.02(100) + b, so the y-intercept is 2200. Therefore, the cost function can be expressed as C(x) = 0.02x + 2200.
b) The graph of this function is a straight line with a positive slope, as shown in the diagram.
c) The slope of the graph is 0.02, which represents the cost per chair.
d) The y-intercept is 2200, which represents the fixed cost of manufacturing the chairs.
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Combine like terms:
5y - 2+4+3y
Answer:
8y+2
Step-by-step explanation:
5y+3y=8y
-2+4=2
Since it's positive two we add them together.
8y+2
Find mangleCHF when mangleBGE = 120°.
mangleCHF = ___°
Answer:
120°
Step-by-step explanation:
AB //CD and EF is transversal. When two parallel lines are cut by a transversal, the alternate exterior angles are congruent.
∠CHF = ∠BGE
= 120°
Calculate the monthly payment and total interest for a loan of $89,000 at 5.7% for 30 years.
Monthly Payment:
Total Interest:
To calculate the monthly payment and total interest for a loan of $89,000 at 5.7% for 30 years, we can use the formula for the monthly payment of a fixed-rate mortgage loan:
P = L[c(1 + c)^n]/[(1 + c)^n - 1]
where P is the monthly payment, L is the loan amount, c is the monthly interest rate (which is the annual interest rate divided by 12), and n is the total number of monthly payments (which is the number of years multiplied by 12).
Plugging in the given values, we have:
L = $89,000
r = 5.7% = 0.057 (monthly interest rate)
n = 30 years * 12 months/year = 360 months
P = ($89,000)[0.057(1 + 0.057)^360]/[(1 + 0.057)^360 - 1]
P = $515.92 (rounded to the nearest cent)
Therefore, the monthly payment is $515.92.
To calculate the total interest, we can subtract the loan amount from the total amount paid over the life of the loan. The total amount paid is equal to the monthly payment times the total number of payments:
Total amount paid = P * n = $515.92 * 360 = $185,731.20
Total interest = Total amount paid - Loan amount = $185,731.20 - $89,000 = $96,731.20
Therefore, the total interest paid over the life of the loan is $96,731.20.
If Q-E-R, and if QE = 8x + 3, QR = 64, and
ER = 4x+1, then find ER.
The measure of ER given by the equation ER = 21
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation be represented as A
Now , the value of A is
Substituting the values in the equation , we get
The measure of QE = 8x + 3 be equation (1)
The measure of ER = 4x + 1 be equation (2)
The measure of QR = 64
Now , QR = QE + ER be equation (3)
Substituting the values of QE and ER in the equation (3) , we get
8x + 3 + 4x + 1 = 64
On simplifying the equation , we get
12x + 4 = 64
Subtracting 4 on both sides of the equation , we get
12x = 60
Divide by 12 on both sides of the equation , we get
x = 5
Substitute the value of x in equation (2) , we get
The measure of ER = 4 ( 5 ) + 1
The measure of ER = 21
Hence , the equation is ER = 21
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Determine if the following equation is linear. If the equation is linear, convert it to standard form: ax+by=c
.
−2x+11(y+x)=2
Step-by-step explanation:
-2x + 11(y+x) = 2
-2x + 11y + 11x = 2
-2x + 11x +11y = 2
9x + 11y = 2
where a = 9, b= 11 and c= 2
factor the expression using GCF 16 X plus 8Y
Factoring the expression 16x+ 8y using GCF would be, 8(2x + 1).
What are GCF and distributive property?The GCF of two or more than two numbers is the highest number that divides the given two numbers completely.
We also know that distributive property states a(b + c) = ab + ac.
Given, An expression 16x+ 8y.
Now, The HCF of 16 and 8 is 8, Because 8 is the highest number that divides 16 and 8 completely.
Therefore, The factored form of the expression 16x+ 8y would be,
8(2x + 1).
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The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method.
x = (y − 6)^2, x = 25; about y = 1
The volume V of the resulting solid is [tex]$\frac{5000 \pi}{3}[/tex]
As per the data given:
The region bounded by the given curves is rotated about the specified axis.
Here we have to determine the volume V of the resulting solid by any method.
The given equations are x = [tex](y -6)^2[/tex], x = 25; about y = 1.
To find the volume of resulting solid we are using cylinder method
To derive the above solution, we need to use the old fashioned quadratic method.
Substitute the values of x then we get
25 = [tex](y-6)^2[/tex]
[tex]\rightarrow 5^2=(y-6)^2[/tex]
y - 6 = 5
y = 6 + 5
y = 11
So, 1 ≤ y ≤ 11
Graph attached at end of the solution
Using cylinder method and integrate along y -axis from y = 1 to y = 11
Volume [tex]}=\int_a^b 2 \pi r h d y[/tex]
Volume [tex]& =2 \pi \int_1^{11}(y-1)\left(25-(y-6)^2\right) d y[/tex]
Volume [tex]=2 \pi \int(y-1)\left(25-y^2-36+12 y\right) d y \\[/tex]
Volume [tex]& =2 \pi \int_1^{11}\left(-y^3+13 y^2-23 y+11\right) d y[/tex]
Volume [tex]=2 \pi\left[-\frac{y}{4}+\frac{13 y}{3}-\frac{23 y}{2}+11 y\right]_1^{11} \\[/tex]
Volume [tex]& =2 \pi \cdot\left(\frac{10043}{12}-\frac{43}{12}\right)=2 \pi\left(\frac{10000}{12}\right)=\frac{5000\pi }{3}[/tex]
Thus the volume of resulting solid is:
[tex]$}=\frac{5000 \pi}{3}[/tex]
Therefore the volume is [tex]$\frac{5000 \pi}{3}[/tex]
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7. Serena bought a scarf for $16.99 and was charged 7.5% tax on her purchase. Which of the
following was the amount of sales tax that Serena had to pay?
(1) $1.19
(3) $1.36
(2) $1.27
(4) $1.51
Answer:
$1.27
Step-by-step explanation:
We know
The tax is 7.5%
7.5% = 0.075
Which of the following was the sales tax Serena had to pay?
We take
16.99 times 0.075 = $1.27
So, the answer is $1.27
The average cost per item, C, of manufacturing a quantity q of cell phones is given by C = (a/q) +b. Find the rate of change of C as q increases. if production increases at a rate of 100 cell phones per week, how fast is the average cost changing?
Answer: The average cost per item is given by the equation C = (a/q) + b. To find the rate of change of C as q increases, we need to take the derivative of this equation with respect to q.
dC/dq = -(a/q^2)
So, the rate of change of C with respect to q is proportional to -1/q^2.
If production increases at a rate of 100 cell phones per week, the rate of change of q is 100. But we need to convert this to units of items per hour or per second, depending on the context.
Let's assume the rate of increase of q is in items per hour. Then the rate of change of C with respect to time t (in hours) is given by:
dC/dt = (dC/dq) * (dq/dt) = (-(a/q^2)) * (100) = -100a/q^2
So, the rate at which the average cost is changing depends on the value of a and q. If we knew the values of a and q, we could calculate the specific rate at which the average cost is changing.
Step-by-step explanation:
We determined that the unconditional probability distribution for Y, the number of defects per yard in a certain fabric, is P(y) = - (1/2)^y+1 y = 0, 1, 2, .... Find the expected number of defects per yard.
The expected number of defects per yard is 1.
The unconditional probability distribution for Y, the number of defects per yard in a certain fabric, is P(y) = - (1/2)^y+1 y = 0, 1, 2, ....
To calculate the expected number of defects per yard, we use the formula E(Y) = ΣyP(y), where y is the number of defects per yard and P(y) is the probability of that number of defects per yard.
We can then substitute the values for P(y) into the formula and sum for all values of y from 0 to ∞. This yields E(Y) = Σy(-(1/2)^y+1) = 1. Thus, the expected number of defects per yard is 1.
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